Einstein summation: power confusion

Question

Assuming $n_i = \sum_i n_i$ (Einstein summation). If I write $n_i^2$, does it mean

1. $(\sum_i n_i)^2$
2. $\sum_i (n_i^2)$

Is there a way to differentiate those two expressions clearly using Eistein summation?

Thank you :)

Answer 1

The Einstein summation convention can't be applied in your example because you only have one index variable ($i$). The convention applies to repeated indices, which is essentially how you avoid ambiguity. So: $$ \sum_i n_i^2 = n_1^2 +n_2^2 +n_3^2+\cdots$$ and $$ n_i^2 = n_i^2 $$ for each $i$ with no summation. However, $$ c^in_i^2 = \sum_i c^i n_i^2 = c^1n_1^2 +c^2n_2^2 + \cdots $$

Conventionally the summation uses a subscript and a superscript as it originally refers to co-variant and contra-variant vectors being summed.

You can also sum over more than one index this way: $$c^ic^jn_in_j^2 = \sum_i \sum_j c^ic^jn_in_j^2 $$